Generalized Indiscernibles as Model-complete Theories
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [kim-kim-scow-2012], [scow-2011]. We understand “theories of indiscernibles” to be special kinds of companionable theories of finite structures, and much of the work in our arguments is carried in the context of the model-companion. Among other things, this approach allows us to prove that the companion of a theory of indiscernibles whose “base” consists of the quantifier-free formulas is necessarily the theory of the Fraïssé limit of a Fraïssé class of linearly ordered finite structures (where the linear order will be at least quantifier-free definable). We also provide streamlined arguments for the result of [kpt-2005] identifying extremely amenable groups with the automorphism groups of limits of Ramsey classes.
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [kim-kim-scow-2012], [scow-2011]. We take “theories of indiscernibles” to be special kinds of companionable Robinson theories in the sense of [hru-1997] that also have a certain unconstrained modeling property (UMP), which formalizes the intuitive notion, “what is needed to run a compactness argument that generates -indiscernibles in a model.” We show that this unconstrained modeling property is equivalent to a structural Ramsey property in the class of finite models of , generalizing certain facts proved in [kim-kim-scow-2012], [scow-2011].
This approach allows us to prove, among other things, that the companion of a theory of indiscernibles whose “base” consists of the quantifier-free formulas is necessarily the theory of the Fraïssé limit of a Fraïssé class of linearly ordered finite structures (recapturing a result that was also proved, more or less, in [kpt-2005] by different means). Our model-theoretic framework also allows us to restate and streamline the arguments of [kpt-2005] – demonstrating the equivalence of the Ramsey property and the extreme amenability of the automorphism group of the Fraïssé limit – in more clearly model-theoretic terms; in particular, for the sufficiency of extreme amenability, we really only to consider the action of the automorphism group on certain type spaces (Stone spaces).
In this article, we present only a few simple examples to go with the technology – leaving applications to a forthcoming companion paper. In that paper, taking [scow-2011] as inspiration, we mainly consider characterizations of classes of theories in the classification-theoretic hierarchy (NIP theories, NSOP theories, and so on) in terms of “collapsing” theories of indiscernibles to reduces.
In this section, we collect almost all of the important definitions around “theories of indiscernibles.” The main concepts are Robinson theories and their model-companions, the Ramsey property and the modeling property for Robinson theories. In the next section, we establish how these ideas fit together.
1.1 Robinson theories
Let be some countable relational language. (Throughout, “no function symbols and only finitely many constant symbols” would also do.)
Wherever it appears, denotes a set of -formulas which contains all of the quantifier-free -formulas, is closed under boolean combinations, under taking subformulas, and under substitutions of free variables. Under these conditions, is called a base.
When is given, we define
Note that may contain sentences.
For -structures , a map is a -embedding if for all and , .
An -theory is -universal if .
Definition 1.2 (Robinson theories).
Let be a countable relational language, and let be a base. We specify a Robinson theory over to be a -universal theory such that:
For any and any , there is a finite -submodel such that .
For every , the set of -complete -types that are consistent with is finite.
(JEP) For all , there are and -embeddings , .
(AP) For all and -embeddings , , there are and -embeddings , , such that .
A model will be called existentially closed (e.c.) if for any -embedding , any , and any ,
If is precisely the theory of e.c. models of , then is called a -model-companion of (provided it is also a companion of with respect to -embeddings). In fact, we can axiomatize more explicitly when it exists. For a finite model , , and an enumeration of with as an initial segment, let be the complete -type of , and let be the sentence ; let be the set of all such sentences. We will see now that a Robinson theory always has a -model-compantion.
Let . Then is e.c. if and only if .
For every model , there is an e.c. model such that .
For a model , let be the following set of sentences (in the expansion of in which every element of is named as a new constant symbol):
where the -formula is just as defined above. From the assumptions that the class of models of has AP and JEP with respect to -embeddings, one finds that has a model, say . Clearly, , and whenever , is finite, and .
Now, as usual, we think of as an operator on models of . We construct a chain of models
(the chain being of order-type ) by taking and, for , . From the assumptions that is closed under boolean combinations and under taking subformulas, it is not hard to see that for all and , where . Moreover, , so is existentially closed. ∎
We also note that the class of models of is a concrete category with -embeddings for morphisms. When this understanding is in play, and when are models of , we write for the set of -embeddings . We may write if there is some risk of ambiguity.
We state a few satisfying facts about Robinson theories in our sense. Their proofs are quite routine, so we omit them.
Assume is a Robinson theory. For every finite model , there is a sentence such that for all , if and only if there is a -embedding .
Let be a Robinson theory over . For every formula , there are and such that
Let be a Robinson theory over . For any model , is consistent.
Our last fact is an immediate consequence of the fact that the class of models of has JEP and AP with respect to -embeddings.
Let be a Robinson theory over . Then is the -model-completion of , and it is a complete theory.
1.2 Modeling and the Ramsey property
We now establish the two properties that could reasonably make a Robinson theory into a theory of indiscernibles. The first of these – the Ramsey property – is essentially a combinatorial partition property of the class of finite models of a Robinson theory . The second – the modeling property – is intended to be just what is needed in order to use the Compactness Theorem to generate an indiscernible model of , which we call an indiscernible -picture, inside a model of a given -theory . First, we need to fix a relatively weak notion of embedding that is more appropriate for accomodating the Ramsey property. (We will also see later in the article that a certain stronger Ramsey property with “standard” -embeddings is not possible.)
Suppose and are -structures. An -embedding consists of a system where for each , is a finite -sub-model of such that , and is a -embedding; we also require that there is a (usually unmentioned) function such that for all . (The stands for “ocalized embedding.”)
Now, suppose , are -embeddings. We define an -embedding as follows: For each , we set and .
Definition 1.10 (Ramsey property).
Let be a Robinson theory over in . We say that has the Ramsey property if the following holds:
Let , and let be a finite model of . For some , let be a finite coloring. Then there is an -embedding such that is constant on for every .
In order to state the modeling property, we must first specify what we mean by a -picture. It is also essential to provide some notion of EM-type for models of ; for this we define the notion of -template. The forms of our definitions here are loosely inspired by the notion of coherent sequences in [hyttinen-2000].
Let be a Robinson theory over in . Let be some language, and let be an -structure.
A -picture in is a pair where and is a one-to-one function.
Let be a -picture in , and let . We say that is -indiscernible if for all and all ,
A -template in is a pair where and is a map satisfying:
respects permutations of coordinates:
For any and , if , then .
For all ,
(Here, indicates a quantifier-free-complete type in the language of equality.)
Let be a -picture in , and let be a -template in . We say that is patterned on if there is a family of patterning maps – meaning that for each , is an -embedding such that
for all , , and .
Definition 1.12 (Modeling property).
Let be a Robinson theory over in . Let be some language, and let be a complete -theory. We say that has the modeling property with respect to if the following holds:
Suppose is a -pattern in , where . Then there are an and an indiscernible -picture in patterned on .
We say that has the unconstrained modeling property (UMP) if it has the modeling property with respect to every complete theory in any countable language.
The terminology “modeling property” is, to the best of our knowledge, due to L. Scow.
2 Results on modeling and the Ramsey property
Now that we have established the relevant definitions, we will determine in this section just how they fit together. In the first subsection, we show that the Ramsey property and the unconstrained model property (together with “fintie rigidity”) are the same thing. In much of the literature on structural Ramsey theory, only Fraïssé classes of linearly ordered finite structures are considered; to some extent, this appears to be just a convenient way of ensuring finite-rigidity. (In general, rigidity does not imply that there is a uniformly definable linear order in a class of finite structures.) As we show in the succeeding subsection, this not in fact just a convenience – any theory of indiscernibles with base has a -definable linear order which is “almost dense.”
2.1 Basic equivalences
The proof of the next lemma was suggested by L. Scow.
Let be a Robinson theory over in . If has the Ramsey property, then has the following finitary Ramsey property:
Let be finite, and let . Then there is a finite model such that for any coloring , there is an such that for all . (That is, is constant on .)
In this proof, it will be convenient to fix a listing of all finite models of up to isomorphism. (This is possible because of the finiteness of for every .)
Let be finite models, and (w.l.o.g.) assume that is nonempty. Let , and towards a contradiction, suppose that for every finite , there is a coloring such that for every , is not constant on . Fix an order of . For each finite , we define an expanded structure as follows222The notation here is so preposterously awful, we couldn’t resist: Cf. for a complex number .:
has an additional sort (in addition to the home-sort ), a new function symbol , and new constant symbols of sort .
, , and for each
For , if there is an such that for each , then . Otherwise, .
Clearly, if are finite and is a -embedding, then extends uniquely to an -embedding . Assume , and let be a formula such that if and , then . (This is possible because is finite for every .) Define similarly. Let be the sentence,
Hence for all finite .
We may then choose a sequence of finite models () such that for all , , and for all , , where for each , is a sentence of such that is equivalent to containing an isomorphic copy of is a -substructure. Let be a non-principal ultrafilter on . Clearly,
and for all . Now, let such that , and let be a coloring extending such that for any finite satisfying , there is a -embedding such that for all . (This is possible just by compactness.)
As has the Ramsey property, there is an -embedding such that is constant on for all . We may assume that appears as a -substructure of , and in this case, is constant on . By construction, there is a -embedding such that for all . It follows that , a contradiction. This completes the proof. ∎
Let be a Robinson theory over in . If has the finitary Ramsey property, then has the Ramsey property.
Let , finite, and be given. We must show that there is an -embedding such that is constant on whenever . By definition of a Robinson theory, for each , let be an finite model of containing . By the finitary Ramsey property, for each , there is a finite model extending such that for any coloring , there is a -embedding such that for all . Since is existentially closed, we may assume that is a -substructure of , so the mappings suffice for the required -embedding. ∎
Let be a Robinson theory over in . Then has the Ramsey property if and only if has the finitary Ramsey property.
Lemma 2.4 (Rigidity).
Let be a Robinson theory over in . If has the Ramsey property, then is finitely-rigid: For all finite, .
Towards a contradiction, suppose is finite, and let be non-trivial. (Obviously, must be an automorphism of .) Let , and let such that for any -embedding , . Now, suppose is an -embedding such that is constant on for every . Fix an embedding and let . For , are distinct, and , a contradiction. ∎
Let be a Robinson theory over in . Then has the Ramsey property if and only if is finitely-rigid and has the unconstrained modeling property.
Assume is finitely-rigid and has UMP. Let be an e.c. model of , and let be a finite model of . Let be a coloring. We define a language with new -ary predicates , where . Let be the -expansion of with the following interpretations: Let ; if with and , then . By finite-rigidity, this is well-defined. Let , and consider the -template in where is the identity map . By UMP, let be an indiscernible -picture in some patterned on , and let , be the family of patterning maps. Fix
We claim that is constant on for every . Let . Then, let be enumerations of and , respectively, such that . By indiscernibility, , so there is a (unique) such that . By definition of patterning and our choice ,
and it follows that . Thus, witnesses the Ramsey property requirement given by . ∎
Proof: “only if”.
Let be a complete -theory, and let be a -pattern in , where is an e.c. model of . Let and . By Lemma 2.4, is finitely-rigid. Let represent all -complete -types over realized in , and for each , let . Without loss of generality, we assume that each enumerates a finite model of – if not we can extend them to finite models and make very minor adjustments to the argument that follows.
Construction: Define as follows:
By the Ramsey property, let be an -embedding such that is constant on for all .
For , given , we define as follows:
By the Ramsey property, let be a -embedding such that is constant on .
Finally, we set
For each , and ,
Note that if is some -embedding, we can start the above construction with in place of , and we would denote this by Construction Now, let () be such that and .
Let be the result of Construction;
Given , let be the result of Construction
In the remainder of the argument, we use the forgoing construction to run a compactness argument that actually generates the required indiscernible -picture. Let be the language with two sorts and , all of the symbols of on the sort , all the symbols of on , constant symbols on for each , and a function symbol . We now define theory that – just to give it a convenient name – we call the Scow-theory of in – it is very similar to a certain theory defined in [scow-2011]. Let be the theory asserting the following:
“ is one-to-one.”
For each , , and ,
where and .
For each and ,
where is the set of --types such that for some , and is the set of functions defined as follows: Let be the smallest number such that ; then where and .
From our prior constructions, it’s not hard to see that this Scow-theory is finitely satisfiable, and in a model , the substructure on is isomorphic to , and the interpretation yields an indiscernible -picture patterned on with ’s as patterning maps. ∎
With these equivalences in place, it’s now natural to define (finally!) what we mean by a “theory of indiscernibles”:
A theory of (generalized) indiscernibles is a Robinson theory that has the Ramsey property.
2.2 Rigidity and order
As promised, in this last subsection, we investigate the finite-rigidity condition in some more depth. In particular, we will see that a theory of indiscernibles must have a 0-definable linear order – indeed, a -definable linear order – so adding a linear order to the language need not provide any new powers. In a rather different manner, this fact was first proved in [kpt-2005] in the quantifier-free case and only recovering a 0-definable linear order. Our result is slightly finer, and our demonstration differs enough, we think, to be interesting in itself.
Definition 2.7 (Irreflexive types).
Let . We say that is irreflexive if .
Let be a Robinson theory, and suppose has the modeling property with respect to an -theory whose models are (expansions of) infinite linear orders. Then for any irreflexive 2-type ,
We prove the lemma with , but this imposes no real loss of generality. With as described, suppose is consistent. Then there are an e.c. model of and such that and . Let be any -template with . By the modeling property with respect to DLO, let , and let be an indiscernible -picture in patterned on . Without loss of generality, suppose ; then since , it must be that as well, which is impossible. Thus,